Given a $n\times n$ integer matrix, what is the best row reduction that can be found using only integer row operations of the form:
- an integer multiple of row $i$ can be added to row $j$
- row i can be multiplied by $\pm 1$
- row $i$ can be swapped with row $j$
It is possible to reduce the matrix into a triangular form. Does the triangular matrix preserve any information from the original matrix? Do the values of the triangular matrix contain any information about the original matrix?
Is there something analogous to the Smith normal form (but presumably less reduced somehow) that exists using only row operations?